The "living" term is mostly to distinguish it from traditional classic convention games.
Normally, in a home game, you create a character and play with a group. You keep the character from session to session, advancing him as you choose according to that game's rules.
Classically, however, if you go to a game convention and play one of the offered games there, you sit down, choose one of the pre-generated characters. You play the game, then discard the character at the end of the game session. If you go to play another game you start over - there is no continuity between games.
"Living" game campaigns try to bring a little of the "home game" experience to convention play. You create a character according to the campaign rules and bring it to the table, You play, and at the end of the game session you are given documentation on what your character achieved and obtained. You can then bring the same character to other games in that campaign, and advance that character's development as you choose.
-k